The hyperbolic-geometry tag has no usage guidance.

**8**

votes

**1**answer

220 views

### Is the group of integer points of ${\rm SO}(n,1)$ maximal?

That is, is it true that there does not exist a lattice in $G = {\rm SO}(n,1)$ which contains the group of integer points of $G$ as a proper subgroup (obviously then of finite index)? if such a ...

**5**

votes

**2**answers

405 views

### injectivity radius of hyperbolic surface

Given a positive real number $l$. Does there exist a closed hyperbolic surface $X$ so that injectivity radius not less than $l$?

**6**

votes

**2**answers

400 views

### Do different Dehn fillings produce homeomorphic 3-manifolds ?

Hi, everyone. I am interested in the dehn filling and Hyperbolic 3-manifold.
Suppose M be an orientable compact 3-manifold with one torus boundary and int(M) admit a
hyperbolic structure. ...

**6**

votes

**1**answer

365 views

### When are isometry groups of hyperbolic 3-manifolds finite?

If $M$ is a finite volume hyperbolic 3-manifold, then its isometry group is finite. I believe this is also true for geometrically finite 3-manifolds. What is the most general condition on a hyperbolic ...

**17**

votes

**2**answers

915 views

### How does hyperbolicity of space time affect our lives?

My main research has been in hyperbolic geometry and geometric group theory. I always thought that the only real "application" of my work was that the universe is a 3-manifold.
But recently I found ...

**6**

votes

**0**answers

323 views

### A conjecture of Thurston and possibly Weeks too

What is the status of the following conjecture:
"... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...

**0**

votes

**1**answer

276 views

### Siegel set in SO(n,1) modulo integer points?

I wonder what is known about a fundamental region for SO($n,1$) modulo its integer points? is there only one cusp? and if one writes a Siegel set in the form of
$K A_\tau N_c$, where $N_c$ is compact ...

**19**

votes

**2**answers

303 views

### Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds

Thurston's Hyperbolic Dehn Surgery Theorem says that all but finitely many fillings of a cusp of a hyperbolic 3-manifold result in hyperbolic manifolds that are deformations of the original manifold. ...

**7**

votes

**1**answer

407 views

### Hyperbolic 3-manifolds with no geometrically finite structure

Does there exist a compact hyperbolic 3-manifold $M$ that is not diffeomorphic to a geometrically finite hyperbolic manifold? If yes, can such $M$ have incompressible boundary?
I think the answer ...

**6**

votes

**2**answers

354 views

### Hyperbolic Brunnian links and rectangular cusp shapes

My question is as follows.
Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components?
Here is what I mean:
The Borromean rings form a famous link $B$ (a smooth ...

**2**

votes

**1**answer

183 views

### Can the finiteness of a Burnside group with two generators be checked algorithmically by using Fuchsian von Dyck groups?

BIG EDIT of the previous question "Coverings of the free Burnside groups", never answered.
In the paper http://link.springer.com/article/10.1007/BF00046586 (last section) there is an interesting ...

**3**

votes

**0**answers

145 views

### reference request: “p-adic” presentation of surfaces

On several occasions I heart about the following result:
For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...

**4**

votes

**1**answer

223 views

### Flows in word-hyperbolic groups

I was wondering if there is a good notion of flows in word-hyperbolic groups (maybe I should say flows in the Cayley graphs of word-hyperbolic groups).
More precisely, I wonder if there is an ...

**6**

votes

**1**answer

406 views

### If rational points are like entire curves, then what do algebraic points correspond to

I read somewhere that if $X$ is a projective variety of general type over a number field $K$, then rational points are an analogue of entire curves $\mathbf{C}\to X^{an}$ (with $X^{an}$ the ...

**8**

votes

**1**answer

351 views

### Geodesic cuffs of pairs of pants in a hyperbolic manifold- why are they disjoint?

I'm trying to understand Kahn-Markovic's celebrated Immersing almost geodesic surfaces in a closed hyperbolic three manifold. There is something probably quite basic which I can't figure out.
We have ...

**6**

votes

**0**answers

609 views

### Closed geodesics on a closed, negatively curved Riemannian manifold

I have been searching for a while for a proof of the following fact: For a closed Riemannian manifold, all of whose sectional curvatures are negative, each free homotopy class of loops contains a ...

**6**

votes

**1**answer

365 views

### Previous work on this generalization of continued fractions?

The 2x2 matrix representation of a continued fraction makes it clear that we're multiplying together a bunch of group elements. Inversion is essentially freely adjoining a generator to a Coxeter ...

**2**

votes

**1**answer

243 views

### growth of energy of eigenfunctions on hyperbolic surface

I am looking to the behavior of eigenfunctions associated to small eigenvalues on a degenerating hyperbolic surface.
Let $(\Sigma_n, h_n)$ a sequence of compact surfaces with area equal to $1$ and ...

**2**

votes

**1**answer

182 views

### Cut and Project Sets Using Hyperbolic Space

One strategy for creating aperiodic sets in $\mathbf{R}$ is to take a line $L$ of irrational slope in $\mathbf{R}^2$ along with a compact window $W \subset \mathbf{R}$ which is thought of as a subset ...

**4**

votes

**1**answer

227 views

### Bisectors in symmetric spaces

In William Goldman's book Complex Hyperbolic Geometry, bisector hypersurfaces play an important role. Given two points $x,y$, the bisector is the set of points equidistant from $x$ and from $y$. Do ...

**10**

votes

**2**answers

424 views

### Some mid-sized ¿hyperbolic? manifolds and SnapPea

I've recently come across some mid-sized 3-manifolds that I think are likely hyperbolic, but SnapPea has some trouble with them. This is related to my previous question
can you fool SnapPea?
but in ...

**6**

votes

**1**answer

248 views

### Can bilipschitz models of hyperbolic 3-manifolds be made effective?

In their proof of the Ending Lamination Conjecture, Brock, Canary, and Minsky prove existence of bilipschitz models of hyperbolic 3-manifolds (homeomorphic to a surface times $\mathbb{R}$) depending ...

**2**

votes

**0**answers

111 views

### Local curvature in a Cayley complex

I'm curious about non-shperical regions in the Cayley complex of a hyperbolic group $G = < X : R \>$ that one might reasonably consider "curved", but I haven't found much discussion of local ...

**2**

votes

**2**answers

565 views

### two curves filling a surface

Let $S$ be a closed surface of genus $g \geq 2$. There exist two simple closed curves filling $S$?
Definitions:
Two closed curves $\alpha, \beta$ fill $S$ if they have minimal intersection and $S ...

**10**

votes

**1**answer

313 views

### Examples of 3-manifolds with RFRS fundamental group

I'm wondering if anyone knows how to construct hyperbolic 3-manifolds whose fundamental group is RFRS. Clearly the recent work of Agol, Wise, etc. says that such manifolds are abundant, and in ...

**5**

votes

**4**answers

574 views

### Synthetic approach to hyperbolic geometry?

Hello,
I am looking for a source that discusses and teaches hyperbolic geometry from a synthetic approach (As opposed to the common analytinc approach in the poincare disk). I am looking for ...

**1**

vote

**2**answers

288 views

### Combination theorems for discrete subgroups of isometry groups

Maskit's combination theorem says: if $M=M_1\cup_\Sigma M_2$ is a union of hyperbolic 3-manifolds $M_1=\Gamma_1\backslash H^3, M_2=\Gamma_2\backslash H^3$ along a surface $\Sigma$, and if the limit ...

**2**

votes

**2**answers

644 views

### Dilogarithm, tetrahedrons, and hyperbolic space

The Bloch-Wigner function $D(z)$, gives the volume of a ideal tetrahedron in hyperbolic space $\mathbb H3$, where z is the cross-ratio (z1,z2,z3,z4) parametrizing the tetrahedron in $\mathbb CP1$.
...

**2**

votes

**2**answers

338 views

### Surface of a Ideal Tetrahedron in Hyperbolic Space H3

The hyperbolic space $\mathbb{H}^3$, has a boundary $\mathbb{CP}^1$.
A ideal tetrahedron in $\mathbb{H}^3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb{CP}^1$.
The four ...

**8**

votes

**0**answers

240 views

### Embeddings of hyperbolic $n$-manifolds in $R^{n+2}$

Is there any example of a compact manifold $M$ of dimension $n>10000$
such that
$M$ admits an embedding into $\mathbb R^{n+2}$,
$M$ is hyperbolic; i.e., it admits a Riemannian metric with
...

**16**

votes

**2**answers

863 views

### Maximum of a function of one variable

Let $D$ be a circular quadrilateral (that is a Jordan region whose boundary consists of 4 arcs
of circles all orthogonal to the unit circle) whose interior angles are all equal to 0, the vertices lie ...

**8**

votes

**2**answers

489 views

### Pseudo-Anosov map, Heegaard splitting, hyperbolic 3-manfold

Hi, I am interested in the relationship between the pseudo-anosov map and volume of the hyperbolic 3-manifold.
Assume $H_{1}$ and $H_{2}$ are two handlebodies with $\partial H_{1}=\partial H_{2}=S$. ...

**6**

votes

**3**answers

357 views

### Are negatively pinched manifold locally conformally flat?

One knows that hyperbolic manifolds are locally conformally flat.
How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy:
$$
-\Lambda \le K \le -\lambda$$
for ...

**1**

vote

**0**answers

166 views

### Bounds on norm of harmonic function on degenerating hyperbolic surface

Suppose I have a Riemann surface with a shrinking geodesic which is degenerating towards a surface with a cusp, and I consider a neighborhood of the shrinking geodesic, $C_[a,b] = [a,b]\times ...

**1**

vote

**1**answer

83 views

### Log-nonexpansive functions: terminology and references

During my recent work in the optimization of positive valued functions, the following class of functions proved to be exceptionally important.
(Defn.). Let $h: (0,\infty) \to (0,\infty)$ be ...

**1**

vote

**2**answers

449 views

### Complete metric on a Riemann surface with punctures

If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric?
I know that in this case the universal cover is the hyperbolic plane ...

**1**

vote

**2**answers

309 views

### Volume of a Riemannian manifold and its relation to fundamental group

I am reading a book (Mapping Class Group by Farb and Margalit) and it says (in a proof of one theorem):
If $S$ admits a hyperbolic metric (they define such a surface to be of finite area and ...

**12**

votes

**1**answer

595 views

### Comparing layered triangulations of 3-manifolds which fiber over the circle.

I am sorry but I am reposting this question because I wasn't logged in when I first asked it.
Ian Agol has produced a method to build an ideal layered triangulation of a hyperbolic 3-manifold which ...

**0**

votes

**0**answers

197 views

### Limit sets of Fuchsian groups and relation between lifts to $H$ of homotopic maps between hyperbolic Riemann surfaces

Let $f,g : X \to Y$ be homotopic (quasiconformal) maps between hyperbolic Riemann surfaces $X,Y$. Consider their (unique) lifts $\tilde{f},\tilde{g}: H\to H$ , that fix $0,1,\infty $. My question is : ...

**0**

votes

**0**answers

198 views

### Quick references/sources for the hyperbolic Riemann Surfaces with boundary

Hello,
Here I am asking for a reference for the universal cover of hyperbolic Riemann surfaces with geodesic boundaries. For example, I want to know how the universal cover/fundamental domain of ...

**16**

votes

**0**answers

438 views

### Actions on ℍⁿ generated by torsion elements

Let $n$ be a large integer.
I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order.
Or equivalently, ...

**4**

votes

**1**answer

333 views

### Volume of a geodesic simplex on a manifold of non-positive curvature.

Let $M$ be a simply connected manifold that admits a metric of non-positive curvature. For example take $\mathbb{R}^k\times \mathbb{H}^n$. Take $m+1$ points $x_0$, $x_1, \ldots$ $x_m$, $m>k+1$ and ...

**0**

votes

**1**answer

206 views

### Teichmuller Theory question : Beltrami forms on hyperbolic Riemann surfaces whose lifts are smooth upto the boundary of $\mathbb{D}$

Hello, my question is related to Teichmuller Theory. Let $D$ be the open unit disk and $X=D/{\Gamma}$ be a hyperbolic Riemann surface of the Fuchsian group $\Gamma$. In Teichmuller theory, we have ...

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votes

**2**answers

435 views

### For which surfaces is Penner's conjecture known to be true?

Robert Penner has proven that, if $A=\{a_1,\dots, a_n\}$ and $B=\{b_1,\dots, b_m\}$ are multicurves in a surface $S$ that together fill $S$, then any product of positive powers of Dehn twists along ...

**2**

votes

**1**answer

255 views

### Discreteness of a group of hyperbolic isometries

Referring to A question about hyperbolic double torus, there is non-discrete $\Gamma= \left< a,b,c,d~~|~~[a,b][c,d] \right> \subset PSL_{2}(\mathbb{R})$ where $a,b,c,d$ are hyperbolic elements. ...

**2**

votes

**1**answer

556 views

### A question about hyperbolic double torus

Hi
I have a question about hyperbolic 2-torus, from now on donoted by $\Sigma_{2}$
Actually I've tried to prove that for a group $\Gamma \subset \textrm{Isom}^{+}(\mathbb{H}^{2})$ represented by ...

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votes

**2**answers

852 views

### Tweetable way to see that Willmore energy is Möbius invariant?

Consider a compact orientable Riemannian manifold $M$ (without boundary) isometrically immersed into $\mathbb{R}^3$. The Willmore energy of $M$ is the functional
$$\mathcal{W} = \int_M H^2 dA$$
...

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votes

**3**answers

494 views

### Torsion in cuspidal cohomology

Following Lemma 2.7 from Vogtmann's Rational Homology of Bianchi Groups, I want to define cuspidal cohomology as $$H_{\mathrm{cusp}}(M)=\frac{H_1(M)}{i_*(H_1(\partial M))}$$ where $i:\partial M\to M$ ...

**6**

votes

**2**answers

289 views

### Hyperbolic structures on $S\times\mathbb{R}$

Let $S$ be a closed incompressible surface in a finite volume hyperbolic 3-manifold $M$ without cusps. Let $N$ be the cover of $M$ associated to $\pi_1(S) \subset \pi_1(M)$. The cover $N$ is ...

**2**

votes

**1**answer

172 views

### Embedding Again

Let $S=[(x,y)\in\mathbb{H}^{2}:0< x< 2\pi]$ where $\mathbb{H}^{2}$ is a hyperplane with standard metric. I.e., a strip whose boundary components are geodesics, both approaching a common infinite ...